Euclidean Axioms to Topological Axioms

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Topology and Topological Space

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Euclidean Axioms to Topological Axioms

 

Topology is the abstract version of geometry. All geometrical shapes are topological spaces. We take some basic features of the geometrical spaces basic features and try to find out what will be the consequences of these features – axiom. Then only we can get a picture of geometrical results and their dependence on axioms.

 

Now, the mathematical proof is a watertight argument that begins with information, proceeds with a logical argument, and ends with proof.  So to define a ‘topological space’ we need axioms. We will start with the axioms in Euclidean geometry

Euclidean axioms:

  • If we take two points, rather any two points, a straight line can be drawn.
  • A line, which is terminated can be extended indefinitely

Topology axioms:

Let $X$ be a non-empty set. A set $T$ of subsets $X$ is said to be a topology on $X$ if:

(I) $X$ and the empty set $ϕ$ belongs to $T$
(II) $X$ the union of any finite or infinite number of sets $T$ belongs to $T$ and
(III) $X$ the intersection of any two sets $T$ belongs to $T$

The pair $(X,T)$ is called a topological space.

Now that we have seen both the type of axioms, so we understand that from here our journey to define topology in mathematical terms would be to prove the axioms. Just as in Euclidean space we find axioms and proof, topology also follows the same rule.

 

Next | What is an Open Set & Open Interval? – Topology

 

See Also | More Topology Articles


Further References:

  • Topology – James R. Munkres
  • Topology without tears – Sidney A.Morris

Article Advisor:

  • Richard Sot (PhD. Mathematics, University of Rochester, Rochester, NY)

Website references:

  1. https://mathworld.wolfram.com
  2. https://www.wikipedia.org/
  3. https://math.stackexchange.com/
  4. https://www.researchgate.net/publication/343635292

 

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